In both cases, we use structured 2D-grids \(\mathcal {T}_h\) , \(\mathcal {T}_H\)  with quadrilateral elements on the domain \(\Omega =[0,1]^2\) .
We specify the number of elements by \(n_h \times n_h\)  and \(n_H \times n_H\)  respectively.
We use the interpolation operator from [1]}
and choose the oversampling parameter \(k\)  as the first integer to satisfy \(k > |\log (H)|\) .
For the evaluation of the error estimator \(\eta _{a, \mu }\)  (REF ) we
approximate \(\gamma _k \approx \alpha ^{1/2}C_{\mathcal {I}_H}^{-1}\)  and assume \(C_{\mathcal {I}_H} \approx 1\) . For quadrilateral elements, the overlapping constant can be explicitly computed
by \(C_{k,\text{ovl}}= (2k + 1)^2\) .
Moreover, we approximate \(\alpha \)  and the contrast \(\kappa \)  by replacing \(\mathcal {P}\)  in eq:ellipticity by the training set
\(\mathcal {P}_\textnormal {train}\) .
Thus, while it is not guaranteed that our approximations yield strict upper bounds on the model order reduction errror, the decay
rate and choice of snapshot parameters will not be affected.
